Derivative of x^(lnx) & ln(cos-¹x)

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Homework Help Overview

The discussion revolves around finding the derivatives of the functions y = x^(lnx) and ln(cos⁻¹x). Participants are exploring the differentiation techniques applicable to these expressions.

Discussion Character

  • Mixed

Approaches and Questions Raised

  • Participants discuss the differentiation of y = x^(lnx) and suggest using logarithmic differentiation. Questions arise regarding the origin of ln(cos⁻¹x) and its differentiation.

Discussion Status

Some participants have made progress on the first function, while others are clarifying the second function. There is acknowledgment of differing interpretations and approaches to the problems presented.

Contextual Notes

One participant notes a discrepancy in the derivative of ln(cos⁻¹x), indicating a potential misunderstanding or error in the textbook answer. The discussion also highlights that the two functions are treated as separate questions.

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Homework Statement


y=x^(lnx) i know the derivative of lnx is 1/x but what about x^(lnx)?

ln(cos-¹x)


Homework Equations


N/A


The Attempt at a Solution



ln(cos-¹x) (1/(cos-¹x))(-x/√(1-x²)) -x/(cos-¹x√(1-x²))
The back in the book has the same answer except the -x is a -1 instead.
 
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Where did the cos-1x come from? :confused:

Usually, when you want to differentiate a function of the form f(x)^g(x), it helps to take the logarithm of the function and then find the derivative.

y = x^ln(x)

ln(y) = ln(x^ln(x))...
Can you do it from here?
 
on the ln(cos^-1x) is a different question.
 
Fusilli_Jerry89 said:
on the ln(cos^-1x) is a different question.
Oh, okay. There is a -1 because the derivative of arccos(x) is -1/sqrt(1-x^2).
 
k I got the first one but i don't get the second one.(ln(cos-¹x))
 
k nm I got it thx
 

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